Analysis of fluctuations in the first return times of random walks on regular branched networks

Abstract

The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks, the variance of FRT, Var(FRT), can be expressed as Var(FRT) = 2⟨FRT⟩⟨GFPT⟩ - ⟨FRT⟩ - ⟨FRT⟩, where ⟨·⟩ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then calculate Var(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e., Cayley tree) by using Var(FRT) and its variant as the metric. We find that the results differ from those in such other networks as Sierpinski gaskets, Vicsek fractals, T-graphs, pseudofractal scale-free webs, (u, v) flowers, and fractal and non-fractal scale-free trees.

Exact results for the first-passage properties in a class of fractal networks

“…and replacing T Ω0 (t − 1) and T B (t − 1) from Eqs. ( 42) and (33) in Eq. ( 43), calculating T H2→H1 and T H1→C1 and inserting them into Eq.…”

“…If we fix the target site (also called trap) and average the MFPT over all the possible starting sites x, we obtain the mean trapping time (MTT) T y for trap site y. In general, one can perform different kinds of average according to the physical situation to be described (see e.g., [29][30][31][32][33][34]): being G(Ω, E) the underlying (undirected) graph, with node set Ω and link set E, one can introduce a distribution p x (with p x ≥ 0 and x∈Ω p x = 1), which represents the likelihood that x is the starting node, in such a way that T y = x∈Ω p x T x→y . The most common situations are the uniform one, where each site has the same probability of being selected (i.e., p x = 1/|Ω|), and the steady-state one, where the probability that a node x is selected as starting site is proportional to its degree d x (i.e., p x = dx 2|E| ).…”

Optimal networks measured by global mean first return time

“…The answers for these questions would be helpful for the design and optimization of networks. The MFRT has close connection with the moments of first passage time [46,48], and the first passage time is useful indicator for transport efficiency of networks. The results obtained in this paper would be helpful for understanding the transport properties of the networks.…”

“…Note that the FRT is a random variable, one can analyze the probability distribution of the FRT [39,[42][43][44][45]. One can also analyze the moments of the FRT [46][47][48]. Fortunately, the mean of the FRT (MFRT) can be exactly evaluated by using the Kac lemma [49], and for classical discrete random walks on finite connected graphs G = (V, E) (V = {v 1 , v 2 , • • • , v n }), the MFRT for random walker starting from vertex v i (i = 1, 2, • • • , n) satisfies ( e.g., see [50,51])…”

A vague memory can affect first-return time

“…Consequently, developing agent-walk models with a limited memory is necessary in order to properly investigate the decision-making processes of animals, which combine recursive walks with exploratory behaviors. Previous studies evaluated the first-return times of fractional Brownian motion and long-term correlated data with distribution densities such as power-law, log normal and so on [28][29][30][31]. These long-term correlations can be linked to memory effects [30,31].…”